Panel collapse and its applications

Abstract

We describe a procedure called panel collapse for replacing a CAT(0) cube complex $Ψ$ by a "lower complexity" CAT(0) cube complex $Ψ_\bullet$ whenever $Ψ$ contains a codimension-$2$ hyperplane that is extremal in one of the codimension-$1$ hyperplanes containing it. Although $Ψ_\bullet$ is not in general a subcomplex of $Ψ$, it is a subspace consisting of a subcomplex together with some cubes that sit inside $Ψ$ "diagonally". The hyperplanes of $Ψ_\bullet$ extend to hyperplanes of $Ψ$. Applying this procedure, we prove: if a group $G$ acts cocompactly on a CAT(0) cube complex $Ψ$, then there is a CAT(0) cube complex $Ω$ so that $G$ acts cocompactly on $Ω$ and for each hyperplane $H$ of $Ω$, the stabiliser in $G$ of $H$ acts on $H$ essentially. Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise. Next, we show that the CAT(0) cube complexes constructed by Cashen-Macura can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of $\mathrm{Out}(F_n)$ on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.